We mention that Liouville type theorems in unbounded domains play a
crucial role to obtain a priori \(L^\infty \) -bounds for solutions of semilinear boundary value problems in bounded
domain, see [1]} for the case of positive solutions and [2]}, [3]}, [4]} and [5]} for sign changing solutions and having finite
Morse index. Also, nonlinear Liouville type theorems combined with the degree type arguments, are useful
to obtain the existence of solutions (see for instance [6]}).
